| Abstract: |
| We investigate some analytic questions concerning the solutions to the equations of a model related to the dynamical interaction between a rigid-body and a viscous incompressible fluid. The model is a generalization of Oseen problem related to the ordinary Navier-Stokes equations. Actually, on its surface $\partial B$, the rigid body B is forced by the Newtonian stress tensor, that is the dynamical answer of the incompressible fluid.
We are able to discuss some questions of uniqueness of regular solutions, asymptotic properties of the solutions and a possible structure theorem related to a suitable weak solution.
All the questions are studied by considering the IBVP in a frame which is attached to the rigid body.
This frame from one side simplifies the approach to the existence of solutions, from another side it makes the inconvenient to introduce the velocity of the rigid motion inside the equations as a coefficient.
In its entirety the problem of the existence of regular solutions was solved in 2002, the case of global solutions for small data in 2023, instead the other questions, that are the uniqueness, asymptotic behavior of the kinetic energy and partial regularity of a weak solution, they were open problems until 2025. |
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